A researcher manually runs two OLS regressions to implement 2SLS: first regressing x on instrument z to get x̂, then regressing y on x̂. She reports the standard errors from the second OLS regression as her 2SLS standard errors. What is wrong with this procedure?
AThe point estimate of the causal effect is also biased when 2SLS is run manually
BThe second-stage OLS standard errors are correct only if the first-stage R² exceeds 0.5
CThe standard errors from manual second-stage OLS are incorrect — they ignore the sampling variability introduced in the first stage and will typically be too small
DThe second stage should regress y on the original x, not on x̂
The point estimate is actually correct when 2SLS is run manually — that is why this error is so dangerous. The standard errors, however, are wrong. Manual second-stage OLS treats x̂ as if it were a fixed, known quantity, but x̂ itself was estimated from data in the first stage. Ignoring that estimation uncertainty understates the true standard errors, leading to t-statistics that are too large and confidence intervals that are too narrow — making results appear more precise than they are. Always use dedicated IV/2SLS software routines that compute the correct asymptotic standard errors.
Question 2 Multiple Choice
A researcher reports a first-stage F-statistic of 4.2 when using a single instrument for an endogenous regressor. What is the key concern about the 2SLS estimates?
AThe instrument may violate the exclusion restriction, as indicated by the low F-statistic
BThe instrument is weak — it explains too little variation in x, so 2SLS estimates are severely biased toward OLS with inflated standard errors
CThe overidentification test will necessarily fail with a low first-stage F-statistic
DThe second stage cannot be run if the first-stage F-statistic falls below 10
The first-stage F-statistic tests instrument relevance — how much variation in the endogenous variable x does the instrument explain? With F = 4.2 (well below the rule-of-thumb threshold of 10), the instrument is weak. Weak instruments cause 2SLS to perform poorly in finite samples: estimates are biased toward OLS (which has the original endogeneity problem), and standard errors become unreliable. Note that a low F-statistic does not indicate exclusion restriction violation — that is a separate issue that F-stat cannot detect.
Question 3 True / False
Having more instruments than endogenous variables (overidentification) allows the researcher to fully verify that most instruments satisfy the exclusion restriction via the Hansen-Sargan J-test.
TTrue
FFalse
Answer: False
The J-test provides only a partial check. It tests whether all instruments yield the same coefficient estimate — if one instrument is invalid (correlated with the error), using it alone would produce a different estimate than using the others. A significant J-test flags inconsistency among instruments. However, the test cannot identify which instrument is invalid, and crucially, if all instruments are invalid in the same direction, the J-test may not detect the problem at all. A passing J-test is not proof of validity — it is merely absence of detected inconsistency.
Question 4 True / False
The first stage of 2SLS isolates the exogenous variation in the endogenous variable x by regressing x on the instrument z, producing fitted values x̂ that are uncorrelated with the error term.
TTrue
FFalse
Answer: True
This is the core logic of 2SLS. The endogenous variable x is contaminated — it is correlated with the error term through omitted variables or reverse causality. By regressing x on the exogenous instrument z, the fitted values x̂ contain only the variation in x that is attributable to z. Since z is assumed exogenous (uncorrelated with the error term, by the exclusion restriction), x̂ inherits that exogeneity. The second stage then uses x̂ as if it were a clean, exogenous regressor.
Question 5 Short Answer
Why does 2SLS produce unbiased causal estimates when OLS does not, and what role does the first stage play?
Think about your answer, then reveal below.
Model answer: OLS regresses y on the endogenous x, which is correlated with the error term — so OLS picks up both the causal effect of x and the confounding relationship. The first stage purges the problem by regressing x on the exogenous instrument z; the fitted values x̂ contain only the variation in x driven by z, which is uncorrelated with the error term (by the exclusion restriction). When the second stage regresses y on x̂, it uses only this clean variation, recovering the causal effect of x without confounding. The endogenous component of x is left behind in the first-stage residuals.
The key intuition: z shifts x for reasons unrelated to the omitted confounders. By 'following' only the variation in x caused by z, 2SLS recovers the effect of x on y through a channel that is free of the bias afflicting OLS. This is why instrument relevance (z actually moves x) and the exclusion restriction (z affects y only through x) are both necessary conditions.